Optimally adapted finite elements meshes

Jean-Marie Mirebeau (1)

1 LJLL - Laboratoire Jacques-Louis Lions

Given a function f defined on a bounded bidimensional domain and a number N, we study the properties of the triangulation T_N that minimizes the distance between f and its interpolation on the associated finite element space,over all triangulations of at most N elements.The error is studied in the L^p norm for 1≤ p ≤ ∞ and we consider Lagrange finite elements of arbitrary polynomial order m-1. We establish sharp asymptotic error estimates as N → ∞ when the optimal anisotropic triangulation is used. These estimates involve invariant polynomials applied to the m-th order derivatives of f. In addition, our analysis also provides with practical strategies for designing meshes such that the interpolation error satisfies the optimal estimate up to a fixed multiplicative constant. We partially extend our results to higher dimensions for finite elements on simplicial partitions of a higher-dimensional domain.

Domaines: Analyse numérique [cs.NA]

Jean-Marie Mirebeau. Optimally adapted finite elements meshes. Constructive Approximation, 2010, 32 (2), pp.339-383. ⟨10.1007/s00365-010-9090-y⟩. ⟨hal-00387807v2⟩